JUNE 2018 MATH2069 Page 2
Use a separate book clearly labelled Question 1
1. [20 marks]
i) [5 marks]
a) Find where the curve
r(t) = ti+ t2(j+ k)
intersects the plane
x+ 7y + 8z = 0.
b) Write a parametric vector equation for the tangent line to the curve
at each point of intersection.
c) Find the cosine of the angle between the curve and the normal direc-
tion to the plane at each point of intersection.
ii) [5 marks]
a) Find an equation for the tangent plane to the sphere
x2 + y2 + z2 − 2y − 4z + 2 = 0
at the point (1, 2, 1).
b) Show that the normal to the sphere is perpendicular to the normal
to the paraboloid
3x2 − 2y + 2z2 = 1
at their point of intersection (1, 2, 1).
iii) [5 marks] Let f(x, y) = x2 − 2y3.
a) Find a unit vector in the xy-plane which points in the direction of
greatest increase of f at the point (1, 2).
b) Find the directional derivative of f at the point (1, 2) in the direction
of the vector i − j.
iv) [5 marks] Use the method of Lagrange multipliers (or any other method
that works) to find the maximum and minimum values of the function
xy2 on the circle x2 + y2 = 1.
Please see over . . .
JUNE 2018 MATH2069 Page 3
Use a separate book clearly labelled Question 2
2. [20 marks]
i) [6 marks] Find the volume of the region bounded by the cone
z =
√
3(x2 + y2)
and the sphere
x2 + y2 + z2 = 5.
(Hint: you may use spherical coordinates)
ii) [8 marks] Let F be the vector field
F(x, y, z) = (yzexyz)i+ xzexyzj+ (xyexyz + z)k.
a) Show that F is conservative by computing its curl.
b) Find a scalar potential function φ with the property ∇φ = F.
c) Hence, or otherwise, calculate∫
C
F(r) · dr,
where C is the curve
r(t) = t5i− t3j+ tk, 0 ≤ t ≤ 1.
iii) [6 marks] Use Gauss’ Divergence Theorem (or any other method that
works) to find the flux of the vector field
F(x, y, z) = xi+ ex+zj+ sin(x− y)k,
out of the solid bounded by the paraboloid z = 4 − x2 − y2 and the
xy-plane.
Please see over . . .
JUNE 2018 MATH2069 Page 4
Use a separate book clearly labelled Question 3
3. [20 marks]
i) [4 marks] Let
f(x+ iy) = x5 − iy5
a) Determine the set of points where f is differentiable.
b) Where is f analytic? Give a reason for your answer.
c) Find f ′(x+ iy) where it exists.
ii) [6 marks] Given that the function u : R2 → R defined by
u(x, y) = cosh x cos y − sinhx sin y + x2 − y2.
is harmonic (you do not need to prove this):
a) Find a harmonic conjugate v for u.
b) Let f(x + iy) = u(x, y) + iv(x, y) for all x, y ∈ R, for the function v
found in the previous part. Find f(z) as a function of z alone.
iii) [3 marks] Find all values of the expression
i2−i
in Cartesian form. Which of these is the principal value?
iv) [7 marks] Suppose that
g(z) =
cos(pi
2
z)
(z + 1)(z − 1)2 .
Let Γ denote the circle with centre at 0 and radius 3, traversed in the
anticlockwise direction.
a) Show that the function g has a removable singularity at z = −1.
Find and classify any other singularity of g.
b) Determine the residue of g at z = 1.
c) Hence, or otherwise, calculate the integral∫
Γ
g(z)dz.
Please see over . . .
JUNE 2018 MATH2069 Page 5
Use a separate book clearly labelled Question 4
4. [20 marks]
i) [6 marks] Evaluate the contour integral∫
γ
z¯ + |z|2 dz,
where
a) γ is the straight line from 1− i to 1 + i;
b) γ is the upper semicircle of unit radius and center 0, traversed anti-
clockwise.
ii) [7 marks] Suppose that
f(z) =
−6
(z + 1)(z − 5) .
a) Write down the three (maximal) regions with centre 1 on which f(z)
has a convergent Laurent (or Taylor) expansion in powers of z − 1.
b) Find the Laurent series expansion of f in powers of z − 1 which is
convergent at the point z = 4.
iii) [7 marks] Use complex analysis methods to find
I1 =
∫ ∞
−∞
cosx
x2 − 2x+ 2 dx
and
I2 =
∫ ∞
−∞
sinx
x2 − 2x+ 2 dx .
End of Exam





















































































































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